Dyer, 28, had a successful operation this week to remove extra bone that had grown on his ankle.
"The operation went well," Swansea's head of performance Richard Buchanan told the club's official website.
"The specialists are pleased with the outcome, and it is estimated that Nathan is likely to be out for between seven and nine weeks."
Dyer has made two appearances this season after being on loan at Premier League champions Leicester last term.
He agreed a new four-year Swansea contract in August.